L-function 1-161-161.132-r0-0-0

• Label: 1-161-161.132-r0-0-0
• Degree: $1$
• Conductor: $161$
• Sign: $-0.764 - 0.644i$
• Analytic cond.: $0.747680$
• Root an. cond.: $0.747680$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.654 − 0.755i)2^-s + (−0.841 − 0.540i)3^-s + (−0.142 + 0.989i)4^-s + (0.415 − 0.909i)5^-s + (0.142 + 0.989i)6^-s + (0.841 − 0.540i)8^-s + (0.415 + 0.909i)9^-s + (−0.959 + 0.281i)10^-s + (0.654 − 0.755i)11^-s + (0.654 − 0.755i)12^-s + (0.959 − 0.281i)13^-s + (−0.841 + 0.540i)15^-s + (−0.959 − 0.281i)16^-s + (−0.142 − 0.989i)17^-s + (0.415 − 0.909i)18^-s + (−0.142 + 0.989i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$161$    =    $7 \cdot 23$
Sign:	$-0.764 - 0.644i$
Analytic conductor:	$0.747680$
Root analytic conductor:	$0.747680$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{161} (132, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 161,\ (0:\ ),\ -0.764 - 0.644i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.2180324695 - 0.5966399603i$
$L(1)$	$\approx$	$0.4986271423 - 0.4347501709i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	7	$1$
	23	$1$
good	2	$1 + (-0.654 - 0.755i)T$
	3	$1 + (-0.841 - 0.540i)T$
	5	$1 + (0.415 - 0.909i)T$
	11	$1 + (0.654 - 0.755i)T$
	13	$1 + (0.959 - 0.281i)T$
	17	$1 + (-0.142 - 0.989i)T$
	19	$1 + (-0.142 + 0.989i)T$
	29	$1 + (-0.142 - 0.989i)T$
	31	$1 + (-0.841 + 0.540i)T$

Imaginary part of the first few zeros on the critical line

−27.9892900028774248450203406688, −27.25560394149928335789126798967, −25.98193699697998832301990384861, −25.79251632915681651094925382655, −24.20825922769926711632978129349, −23.33378689915079100041124594065, −22.473322369514329938518938687998, −21.638085326164475092612940421356, −20.22334856974294646621588764962, −18.934096929878654785599243021359, −17.96602515724918005584047893381, −17.37280531895168709356018734976, −16.37559821991271631249676705263, −15.23498972075297812141573070050, −14.66487554684451763101486538361, −13.2445100684186460594795402825, −11.48718822776663156375670719651, −10.6711253127042574272760713899, −9.79214500060649557684495590592, −8.75392032157110504982822932056, −6.97127832985629699023758208718, −6.44377376719076425739155483117, −5.2932657327746562605205797104, −3.88576218001370271057452721725, −1.64871255878768995867177756000, 0.802866696272522296463450144293, 1.88634489203604345665169947988, 3.77566703716867311987496177939, 5.25568326158553464494039557899, 6.47408877675984108838068349951, 7.95083910928779547301472746357, 8.90087031056658487446545686986, 10.09573315013369146658078979317, 11.263454122043851061987087253584, 12.0204995228708000408241993803, 13.05330135058468614934910858605, 13.79244872037721486880691126122, 16.18605138015820998535351231195, 16.60321628825596102455021783879, 17.67770388522213544950904151899, 18.4286122664025951375335889298, 19.439753706000747548209577755969, 20.54392813363357877649857240136, 21.40775516826510842155590746309, 22.41617829856523798837157228761, 23.42799393407636405063552909855, 24.822519445899349920743594757786, 25.19623428275171171283767811782, 26.82728767262370656827077291882, 27.73859367160972513141268030689

Graph of the $Z$-function along the critical line